What do Muslims believe about reincarnation?
In general, the past-life regression business hasn't got much of a future in Muslim territories. Islam has no place for returning in the form of another person after dying. There is no leeway in this teaching.
Islam teaches that the purpose of each life is to encounter Allah--or God--and that no matter how limited the life, it is sufficient to determine that person's ultimate destiny--paradise or perdition.
Islam clearly defines the soul as part of each specially-created individual and holds that nothing can change that. The soul cannot be detached and move on to another body; it cannot engage in upward or downward mobility in the chain of creation; elements of it, such as memory, cannot be transplanted into someone in a later generation.